Optimal Control of Trading Algorithms: A General Impulse Control Approach

We propose a general framework for intra-day trading based on the control of trading algorithms. Given a generic parameterized algorithm, we control the dates (τi)i at which it is launched, the length (δi)i of the trading period and the value of the parameters (Ei)i kept during the time interval [τi, τi + δi[. This gives rise to a non-classical impulse control problem where not only the regime Ei but also the period [τi, τi +δi[ has to be determined by the controller at the impulse time τi. We adapt the weak dynamic programming principle of Bouchard and Touzi (2009) to our context and provide a characterization of the associated value function as a discontinuous viscosity solution of a system of PDEs with appropriate boundary conditions, for which we prove a comparison principle. We also propose a numerical scheme for the resolution of the above system and show that it is convergent. We finally provide an example of application to a problem of optimal stock trading with a non-linear market impact function.